Home page for Math 317/001:

Advanced Calculus of One Variable

Fall Semester 2006


Holger Kley	Time: MWF 10:00--10:50
Hours: Mondays 11--12, 4--5, and by appointment	Place: Engineering E105
Problem Session: Thursdays 10:00--10:50, Engineering E105


Announcements	Description	Schedule	Text	Homework	Grading	Documents	Links

Announcements:


Please look at the Documents section!	12/12
Finals week office hours: Tuesday 3--5, Wednesday 12--1 and 4--6.	12/6
11/13 Office hours: 3:30--4:00 and 5:00--5:30.	11/10
Typo alert!!! For credit problem 1(b), 1(d), and #4 from Section 8.1 of the book had typos. Please see the see corrected version.	11/6
Additional office hour this week only: Thursday 12--1.	10/23
Schedule change, this week only. No office hours Monday 9/4: Labor Day. HW 2 (short assignment) due Wednesday, 9/6. Office hours Tuesday 9/5: 9 am and 5 pm.	8/29

Description:

The official prerequisites for this course are one-variable calculus, meaning M160 and M161. The focus in those courses is on conceptual understanding, problem-solving and computation, and while simple proofs are occasionally carried out, they are not emphasized, and students are not expected to develop skill at constructing proofs.

In M317, on the other hand, the emphasis is almost entirely on proof. Essentially, we will be exploring the foundations of mathematical analysis in one (real) variable, as established by Cauchy, Weierstrass and others in the 19th century. Almost every one of the major theorems on which the techniques of M160/161 rely will be proved in this course. This is a fairly tall order, and as a result, it's usually helpful to have had a few additional math courses before attempting this one, in order to begin developing "mathematical maturity."

Schedule of Lectures:

For the most part, we will be following the text. The titles of the lectures give a theme or focus and are not meant to describe everything which may be covered on any given day. I've added roughly corresponding section number from the text, but this certainly does not mean that I will cover every word in those sections. Since I've undoubtedly misestimated how long it will take to cover certain topics, this schedule is subject to change.

	Date	Lecture	Text
Week 1	Aug. 21	Introduction and warm-up	Preliminaries and Appendix A
	Aug. 22	The real numbers are complete	1.1
	Aug. 23	The Archimedean property and the density of the rationals	1.2
	Aug. 25	The Archimedean property and the density of the rationals	1.2
Week 2	Aug. 28	Mathematical Induction and a few useful identities	1.3
	Aug. 29	Sequences and their limits	2.1
	Aug. 30	Epsilon-N proofs	2.1
	Sep. 1	Sequences and completeness	2.2
Week 3	Sep. 4	Labor Day; no class!
	Sep. 5	Bounded monotone sequences converge	2.3
	Sep. 6	Subsequences	2.4
	Sep. 8	Subsequences	2.4
Week 4	Sep. 11	Continuous functions I: definition via sequences	3.1
	Sep. 12	Continuity and Completeness I: extreme values	3.2
	Sep. 13	Continuity and Completeness II: intermediate values	3.3
	Sep. 15	Continuity and Completeness II: intermediate values	3.3
Week 5	Sep. 18	Uniform continuity	3.4
	Sep. 19	Uniform continuity	3.4
	Sep. 20	Catch-up and review
	Sep. 22	Test 1
Week 6	Sep. 25	Continuous functions II: the epsilon-delta definition	3.5
	Sep. 26	When is the inverse of a continuous function continuous?	3.6
	Sep. 27	When is the inverse of a continuous function continuous?	3.6
	Sep. 29	Derivatives	4.1
Week 7	Oct. 2	Derivatives	4.1
	Oct. 3	The chain rule	4.2
	Oct. 4	The inverse function theorem (in one variable)	4.2
	Oct. 6	The mean value theorem	4.3
Week 8	Oct. 9	The mean value theorem	4.3
	Oct. 10	The Cauchy mean value theorem	4.4
	Oct. 11	Darboux Sums and the definition of the Riemann integral	6.1
	Oct. 13	Darboux Sums and the definition of the Riemann integral	6.1
Week 9	Oct. 16	A criterion for integrability	6.2
	Oct. 17	Properties of the Riemann integral	6.3
	Oct. 18	Properties of the Riemann integral	6.3
	Oct. 20	Continuous functions are integrable	6.4
Week 10	Oct. 23	The first fundamental theorem of calculus	6.5
	Oct. 24	The first fundamental theorem of calculus	6.5
	Oct. 25	Catch-up and review
	Oct. 27	Test 2
Week 11	Oct. 30	The second fundamental theorem of calculus	6.6
	Oct. 31	Common techniques of integration	7.2
	Nov. 1	Taylor Polynomials	8.1
	Nov. 3	Taylor's theorem I: Lagrange form	8.2
Week 12	Nov. 6	Taylor's theorem I: Lagrange form	8.2
	Nov. 7	Taylor series	8.3
	Nov. 8	Taylor's theorem II: Cauchy form	8.5
	Nov. 10	Series	9.1
Week 13	Nov. 13	Series	9.1
	Nov. 14	Sequences of functions: pointwise and uniform convergence	9.2--3
	Nov. 15	Sequences of functions: pointwise and uniform convergence	9.2--3
	Nov. 17	Uniform limits	9.4
Week 14	Nov. 27	Power series	9.5
	Nov. 28	Analyticity	8.6
	Nov. 29	The Weierstrass approximation theorem	8.7
	Dec. 1	A nowhere differentiable, continuous function	9.6
Week 15	Dec. 4	Logs, exponentials, and trig functions	5.1--3
	Dec. 5	Logs, exponentials, and trig functions	5.1--3
	Dec. 6	Review
	Dec. 8	Review
Finals Week	Dec. 14	Final Exam, 1:30--3:30 p.m.

Text:

Fitzpatrick, Patrick M. Advanced Calculus, 2nd ed., Thomson Brooks/Cole, Belmont, 2006. ISBN 0-534-37603-7. The text (affectionately known as Fitzpatrick) is available at the CSU bookstore.

Homework:

Homework will be assigned weekly (except the last week), on Mondays or Tuesdays. Problems from the book will occasionally be supplemented with problems of my own design. Each homework assignment will consist of two kinds of problems:

Discussion problems will form the basis of our Thursday problem session and can be discussed (in detail, if needed) in office hours and (if needed) in class. You are encouraged to work together on discussion problems, and to write down rought drafts of solutions. Discussion problems will not be turned in for a grade.

For-credit problems will consist of problems similar to discussion problems. Work on for-credit problems must be substantially your own. If you are struggling with one of these problems, I may give a hint, but never a complete solution. If you like, you may think of these as take-home tests. Work that you submit for these problems should be neat and carefully done -- you will probably want to rewrite and polish your first attempts at a solution. Please leave sufficient space in the margins and between problems for the grader's comments. I will post solutions to for-credit problems after they are due. Once a solution is posted, late submissions will not be accepted for credit.

HW 1 ** Solutions to HW 1	HW 2 ** Solutions to HW 2	HW 3 ** Solutions to HW 3	HW 4 ** Solutions to HW 4
HW 5 ** Solutions to HW 5	HW 6 ** Solutions to HW 6	HW 7 ** Solutions to HW 7	HW 8 ** Solutions to HW 8
HW9 ** Solutions to HW 9	HW 11 ** Solutions to HW 11	HW12 ** Solutions to HW 12	HW13 ** Solutions to HW 13
HW14 ** Solutions to HW 14

Grading:

Course grades will be computed as follows:

Assignments 1--9, 11--14	3% each
Tests 1 and 2	15% each
Final	31%
Total	100%

Documents:

(For HW assignments and solutions, please see the Homework section above.)
Class Policies	8/21	Sample Test 1	9/19
Sample Test 2	10/24	Sample Final	12/6
Course Themes	12/6	Test 1 Solutions	12/12
Test 2 Solutions	12/12

Links:

If you know of any good resources that should appear here, please let me know.

The cover story "Math will Rock your World" of the 1/23/06 issue of Business Week examines the expanding role of mathematics in the business sector.

The article "Manifold Destiny" by S. Nasar and D. Gruber in the 8/28/06 issue of The New Yorker examines the recent solution of a long-standing mathematical problems, and the controversy that such solutions may bring.

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